Solution of the υ-representability problem on a one-dimensional torus

Sarina M. Sutter; Markus Penz; Michael Ruggenthaler; Robert van Leeuwen; Klaas J.H. Giesbertz

doi:10.1088/1751-8121/ad8a2a

Solution of the υ-representability problem on a one-dimensional torus

Sarina M. Sutter, Markus Penz^*, Michael Ruggenthaler, Robert van Leeuwen, Klaas J.H. Giesbertz

^*Corresponding author for this work

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

We provide a solution to the v-representability problem for a non-relativistic quantum many-particle system on a one-dimensional torus domain in terms of Sobolev spaces and their duals. Any one-particle density that is square-integrable, has a square-integrable weak derivative, and is gapped away from zero can be realized from the solution of a many-particle Schrödinger equation, with or without interactions, by choosing a corresponding external potential. This potential can contain a distributional contribution but still gives rise to a self-adjoint Hamiltonian. Importantly, this allows for a well-defined Kohn-Sham procedure but, on the other hand, invalidates the usual proof of the Hohenberg-Kohn theorem.

Original language	English
Article number	475202
Pages (from-to)	1-23
Number of pages	23
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	57
Issue number	47
Early online date	5 Nov 2024
DOIs	https://doi.org/10.1088/1751-8121/ad8a2a
Publication status	Published - 22 Nov 2024

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© 2024 IOP Publishing Ltd. All rights, including for text and data mining, AI training, and similar technologies, are reserved.

Keywords

density-functional theory
distributional potential
many-particle Schrödinger equation
v-representability

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10.1088/1751-8121/ad8a2a

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title = "Solution of the υ-representability problem on a one-dimensional torus",

abstract = "We provide a solution to the v-representability problem for a non-relativistic quantum many-particle system on a one-dimensional torus domain in terms of Sobolev spaces and their duals. Any one-particle density that is square-integrable, has a square-integrable weak derivative, and is gapped away from zero can be realized from the solution of a many-particle Schr{\"o}dinger equation, with or without interactions, by choosing a corresponding external potential. This potential can contain a distributional contribution but still gives rise to a self-adjoint Hamiltonian. Importantly, this allows for a well-defined Kohn-Sham procedure but, on the other hand, invalidates the usual proof of the Hohenberg-Kohn theorem.",

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AU - Sutter, Sarina M.

AU - Penz, Markus

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AU - Giesbertz, Klaas J.H.

PY - 2024/11/22

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N2 - We provide a solution to the v-representability problem for a non-relativistic quantum many-particle system on a one-dimensional torus domain in terms of Sobolev spaces and their duals. Any one-particle density that is square-integrable, has a square-integrable weak derivative, and is gapped away from zero can be realized from the solution of a many-particle Schrödinger equation, with or without interactions, by choosing a corresponding external potential. This potential can contain a distributional contribution but still gives rise to a self-adjoint Hamiltonian. Importantly, this allows for a well-defined Kohn-Sham procedure but, on the other hand, invalidates the usual proof of the Hohenberg-Kohn theorem.

AB - We provide a solution to the v-representability problem for a non-relativistic quantum many-particle system on a one-dimensional torus domain in terms of Sobolev spaces and their duals. Any one-particle density that is square-integrable, has a square-integrable weak derivative, and is gapped away from zero can be realized from the solution of a many-particle Schrödinger equation, with or without interactions, by choosing a corresponding external potential. This potential can contain a distributional contribution but still gives rise to a self-adjoint Hamiltonian. Importantly, this allows for a well-defined Kohn-Sham procedure but, on the other hand, invalidates the usual proof of the Hohenberg-Kohn theorem.

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Solution of the υ-representability problem on a one-dimensional torus

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